Stability in open strings with broken supersymmetry Herv e - PowerPoint PPT Presentation

Stability in open strings with broken supersymmetry Herv´ e Partouche CNRS and Ecole Polytechnique April 23 2019 Based on work done in collaboration with S. Abel, E. Dudas and D. Lewis [arXiv:1812.09714]. String theory from a worldsheet perspective, GGI, Firenze 1 / 40

Introduction � Important properties of String Theory (dualities, branes,...) have been discovered in presence of exact supersymmetry in flat space. Susy guaranties stability of flat backgrounds from weak to strong coupling. � For Phenomonology and Cosmology, susy must be broken “In a worldsheet perspective”, we work at string weak coupling. � We can start classically with AdS or flat : Perturbative loop corrections cannot make AdS nearly flat. = ⇒ We start from a Minkowski background. 2 / 40

� If “hard” breaking of susy, the susy breaking scale and effective potential are V quantum ∼ M d M = M s = ⇒ in string frame s = ⇒ Minkowski destabilized � If susy spontaneously broken in flat space classically = “No-scale model” : [Cremmer, Ferrara, Kounnas, Nanopoulos,’83] • V classical is positive, with a minimum at 0, and a flat direction parameterized by M , which is a field V quantum ∼ M d , • String loop corrections = ⇒ generically Better, but still too large. We need non-generic No-Scale Models. 3 / 40

� In this talk : We try to improve the quantum stability of flat backgrounds with spontaneously broken susy. • Lower the order of magnitude of the potential at 1-loop This is modest : Higher loops should be included. Their consistent definition must be addressed. • However, the quantum potential may induces instabilities for internal moduli : Tadpoles ? And if not, tachyonic mass terms ? 1-loop is enough to make good improvements about this issue. • We do this in type I string compactified on tori, but this can be more general (heterotic). 4 / 40

� Susy breaking via stringy Scherk-Schwarz mechanism. • In field theory : Refined version of a Kaluza-Klein dimensional reduction of a theory in d + 1 dimensions If there is a symmetry with charges Q in d + 1 dim, we can impose Q -depend boundary conditions Φ( x µ , y + 2 πR ) = e iπQ Φ( x µ , y ) mass = | m + Q m + Q 2 | 2 Φ( x µ , y ) = � Φ m ( x µ ) e i y ⇒ ⇒ = = R R m = ⇒ A multiplet in d + 1 dim with degenerate states have descendent which are not-degenerate. • If Supersymmetry : Q = F is the fermionic number M = 1 = ⇒ super Higgs 2 R • Generalized in closed string theory [Rohm,’84][Ferrara, Kounnas, Porrati,’88] and in open string theory [Blum, Dienes,’97][Antoniadis, Dudas, Sagnotti,’98] 5 / 40

� Compute the 1-loop effective potential V 1-loop = − M d s 2(2 π ) d ( T + K + A + M ) , � dτ 1 dτ 2 Str q L 0 − 1 L 0 − 1 ˜ 2 ¯ where T = q 2 1+ d F τ 2 2 � + ∞ dτ 2 Str Ω q L 0 − 1 L 0 − 1 ˜ 2 ¯ K = q 2 1+ d 0 τ 2 2 � + ∞ dτ 2 2 ( L 0 − 1 1 2 ) A = Str q 1+ d 0 τ 2 2 � + ∞ dτ 2 2 ( L 0 − 1 1 2 ) M = Str Ω q 1+ d 0 τ 2 2 Str e − πτ 2 M 2 � V ∼ = ⇒ The dominant contribution arises from the lightest states. 6 / 40

Suppose the classical background is such that there is no 1 mass scale between 0 and the susy breaking scale M = 2 R ——— cM s : large Higgs or string scale M s ——— M : towers of Kaluza-Klein modes of masses ∝ M ——— 0 : n B massless bosons and n F massless fermions = ⇒ In string frame, the 1-loop effective potential is dominated by the KK modes V 1-loop = ( n F − n B ) ξ M d + O � 2 e − cM s /M � d ( cM s M ) , where ξ > 0 7 / 40

V 1-loop = ( n F − n B ) ξ M d + O � 2 e − cM s /M � d ( cM s M ) The exponential terms are negligible even for moderate M cM s ∼ M Planck E.g. : For 10 � 2 e − cM s /M � d < 10 − 120 M 4 O we have ( cM s M ) Planck M < 10 − 3 M Planck when √ R > 10 2 ≫ Hagedorn radius R H = NB : = ⇒ 2 /M s , = ⇒ No “Hagedorn-like phase transition” (no tree level tachyon). 8 / 40

• Deform slightly the previous background i.e. switch on small moduli deformations collectively denoted “ a ” ——— cM s : large Higgs or string scale ——— M : towers of KK modes of masses ∝ M ——— aM s : some of the n B + n F states get a Higgs mass aM s ——— 0 • n B ( a ) and n F ( a ) interpolate between different integer values , reached in distinct regions in moduli space. = ⇒ Expand them in “ a ” to find V 1-loop around the initial background. 9 / 40

� Because we compactify on a torus ( N = 4 in 4 dim), all moduli are Wilson lines (WL) : a =0 + M d � � � Q r a I � V 1-loop = V 1-loop r + · · · � r,I massless their KK spectrum modes • a I r is the WL along the internal circle I of the r -th Cartan U (1). • Q r is the charge of the massless spectrum (and Kaluza-Klein towers). • combining states Q r and − Q r = ⇒ 0 : No Tadpole All points in moduli space where there is no mass scale between 0 and M are local extrema. 10 / 40

• This is reminiscent of an argument of Ginsparg and Vafa (’87) in the non-susy O (16) × O (16) heterotic compactified on tori : At enhanced gauge symmetry points, U (1) 26 − d → Non-Abelian, there are additional non-Cartan massless states, with non-trivial Q r . ⇒ Q r → − Q r is an exact symmetry (underlying gauge symmetry) = of the partition function at any genus = ⇒ extremum. • In the Scherk-Schwarz case : The non-existence of tadpoles should be exact (including the exponentially suppressed terms) and at any genus. But the massless states may not contain gauge bosons. In a non-Cartan vector mutiplet, we can keep massless the fermions and give a mass to the bosons. So U (1)’s are still allowed, with charged fermions. 11 / 40

� At quadratic order [Kounnas, H.P,’16][Coudarchet, H.P.,’18] � � 2 + · · · M d + M d � � Q 2 � Q 2 � a I � � � V 1-loop = ξ n F − n B r − r r massless massless their KK bosons fermions modes = ⇒ The higher V 1-loop is, the more tachyonic it is. � We are interested in models where n F = n B and tachyon free at 1-loop to preserve flatness of spacetime (at this order). [Abel, Dienes, Mavroudi,’15][Kounnas, H.P.,’15][Florakis, Rozos,’16] = “Super No-scale Models in String Theory” : The no-scale structure exact at tree level is preserved at 1-loop, up to exponentially suppressed terms i.e. the 1-loop potential is locally positive, with minimum at 0, and with a flat direction M . NB: n F , n B count observable and hidden sectors d.o.f. 12 / 40

� In this talk : We show that tachyon free models with V 1-loop = 0 (or > 0) exist at 1-loop, for d ≤ 5. • In 9 dimensions : We find the models stable with respect to the open string Wilson lines. = ⇒ V 1-loop < 0 = ⇒ runaway of M • In d dimensions : We have - Open string Wilsons lines - Closed string moduli (which also WLs) : NS-NS metric G IJ and RR 2-form C IJ 13 / 40

� Note that in Type II and orientifold theories, there exist non-susy models with V 1-loop = 0 i.e. N F = N B are any mass level ! [Kachru, Kumar, Silverstein,’98] [Harvey,’98] [Shiu, Tye,’98] [Blumenhagen, Gorlich,98] [Angelantonj, Antoniadis, Forger,’99] [Satoh, Sugawara, Wada,’15] However Moduli stability has not been studied ( ⇒ tachyonic at 1-loop). There are no exponentially suppressed terms at 1-loop, but this does not change the fact that V 2-loops has no reason to vanish. [Iengo, Zhu,’00][Aoki, D’Hoker, Phong,’03] When a perturbative heterotic dual is known, it only has n F = n B . [Harvey,’98][Angelantonj, Antoniadis, Forger,’99] 14 / 40

In 9 dimensions � Type I compactified on S 1 ( R 9 ) with Sherk-Schwarz susy breaking • Closed string sector : The states with non-trivial winding n 9 are heavier than the string scale = ⇒ exponentially suppressed → m 9 + F m 9 2 − For n 9 = 0, the momentum R 9 R 9 • Open string sector : 32 D9-branes generate SO (32) on their world volume. Switch on generic Wilson lines (=Coulomb branch) e 2 iπa 1 , e − 2 iπa 1 , e 2 iπa 2 , e − 2 iπa 2 , . . . , e 2 iπa 16 , e − 2 iπa 16 � W = diag � → m 9 + F 2 + a r − a s m 9 momentum − R 9 R 9 (The Chan-Paton charges are absorbed in the WLs : Q r a r → a r ) 15 / 40

� T-duality R 9 → ˜ R 9 = α ′ R 9 yields a geometric picture in X 9 : Type I’, where WLs become positions along ˜ • S 1 ( R 9 ) becomes S 1 ( ˜ R 9 ) / Z 2 i.e. a segment with 2 O8-orientifold X 9 = 0 and ˜ X 9 = π ˜ planes at ˜ R 9 . • The D9-branes become 32 D8 “half”-branes : X 9 = 2 πa r ˜ X 9 = − 2 πa r ˜ 16 at ˜ R 9 and 16 mirror 1 2 -branes at ˜ R 9 . 1 2 -branes and mirrors 1 • 2 -branes can be coincident on an O8-plane, a r = 0 or 1 ⇒ = SO ( p ), p even 2 • Elsewhere, a stack of q 1 2 -branes and the mirror stack = ⇒ U ( q ) 16 / 40

� We look for stable brane configurations. • A sufficient condition for V 1-loop to be extremal with respect to the a r is that there is no mass scale between 0 and M . Thus, we may concentrate on a r = 0 or 1 2 only, i.e. no brane in the bulk. • Moreover, this special case yields massless fermions because m 9 + F 2 + a r − a s = m 9 + 1 2 + 1 2 − 0 can vanish (where m 9 is a winding number in the T-dual picture) i.e. Super-Higgs and Higgs compensate This is a good point to have n F − n B ≥ 0. 17 / 40